As of now, in the PC version of Terraria, you can have one minion, one passive pet, one mountable pet, and one light giving pet out at the same time (ie. Pygmy, Sapling, Bunny Mount, and Wisp in a bottle).How do you get a pet on Terraria?
There are currently 26 pets available for the PC version of Terraria. You can find a pet by getting them as a drop from successfully defeating certain bosses or finding them in certain chests. The Terraria Wiki covers Pets more thoroughly, I recommend you check it out.